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One ninth of 9-factorial numbers.
14

%I #39 Oct 20 2022 03:34:24

%S 1,18,486,17496,787320,42515280,2678462640,192849310080,

%T 15620794116480,1405871470483200,139181275577836800,

%U 15031577762406374400,1758694598201545804800,221595519373394771404800,29915395115408294139648000,4307816896618794356109312000

%N One ninth of 9-factorial numbers.

%C E.g.f. is g.f. for A001019(n-1) (powers of nine).

%H G. C. Greubel, <a href="/A035023/b035023.txt">Table of n, a(n) for n = 1..325</a>

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%F 9*a(n) = (9*n)(!^9) = Product_{j=1..n} 9*j = 9^n*n!.

%F E.g.f.: (-1+1/(1-9*x))/9.

%F D-finite with recurrence: a(n) - 9*n*a(n-1) = 0. - _R. J. Mathar_, Jan 28 2020

%F From _Amiram Eldar_, Jan 08 2022: (Start)

%F Sum_{n>=1} 1/a(n) = 9*(exp(1/9)-1).

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 9*(1-exp(-1/9)). (End)

%F From _G. C. Greubel_, Oct 19 2022: (Start)

%F a(n) = A001019(n-1) * A000142(n). - _G. C. Greubel_, Oct 19 2022

%t With[{nn=20},Rest[CoefficientList[Series[(-1+1/(1-9*x))/9,{x,0,nn}],x] Range[ 0,nn]!]] (* _Harvey P. Dale_, Apr 07 2019 *)

%t Table[9^(n-1)*n!, {n, 40}] (* _G. C. Greubel_, Oct 19 2022 *)

%o (Magma) [9^(n-1)*Factorial(n): n in [1..40]]; // _G. C. Greubel_, Oct 19 2022

%o (SageMath) [9^(n-1)*factorial(n) for n in range(1,40)] # _G. C. Greubel_, Oct 19 2022

%Y Cf. A000142, A007559, A001019, A034171, A035012, A035013.

%Y Cf. A035017, A035018, A035020, A035021, A035022, A045756.

%K easy,nonn

%O 1,2

%A _Wolfdieter Lang_